### Abstract

We consider the finite exceptional group of Lie type G=E_{6} ^{ε}(q) (universal version) with 3|q−ε, where E_{6} ^{+1}(q)=E_{6}(q) and E_{6} ^{−1}(q)=^{2}E_{6}(q). We classify, up to conjugacy, all maximal-proper 3-local subgroups of G, that is, all 3-local M<G which are maximal with respect to inclusion among all proper subgroups of G which are 3-local. To this end, we also determine, up to conjugacy, all elementary-abelian 3-subgroups containing Z(G), all extraspecial subgroups containing Z(G), and all cyclic groups of order 9 containing Z(G). These classifications are an important first step towards a classification of the 3-radical subgroups of G, which play a crucial role in many open conjectures in modular representation theory.

Original language | English |
---|---|

Pages (from-to) | 4020-4039 |

Number of pages | 20 |

Journal | Journal of Pure and Applied Algebra |

Volume | 222 |

Issue number | 12 |

DOIs | |

Publication status | Published - 2018 Dec 1 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

_{6}

*Journal of Pure and Applied Algebra*,

*222*(12), 4020-4039. https://doi.org/10.1016/j.jpaa.2018.02.018

}

_{6}',

*Journal of Pure and Applied Algebra*, vol. 222, no. 12, pp. 4020-4039. https://doi.org/10.1016/j.jpaa.2018.02.018

**Classification of some 3-subgroups of the finite groups of Lie type E _{6} .** / An, Jianbei; Dietrich, Heiko; Huang, Shih-Chang.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Classification of some 3-subgroups of the finite groups of Lie type E6

AU - An, Jianbei

AU - Dietrich, Heiko

AU - Huang, Shih-Chang

PY - 2018/12/1

Y1 - 2018/12/1

N2 - We consider the finite exceptional group of Lie type G=E6 ε(q) (universal version) with 3|q−ε, where E6 +1(q)=E6(q) and E6 −1(q)=2E6(q). We classify, up to conjugacy, all maximal-proper 3-local subgroups of G, that is, all 3-local M<G which are maximal with respect to inclusion among all proper subgroups of G which are 3-local. To this end, we also determine, up to conjugacy, all elementary-abelian 3-subgroups containing Z(G), all extraspecial subgroups containing Z(G), and all cyclic groups of order 9 containing Z(G). These classifications are an important first step towards a classification of the 3-radical subgroups of G, which play a crucial role in many open conjectures in modular representation theory.

AB - We consider the finite exceptional group of Lie type G=E6 ε(q) (universal version) with 3|q−ε, where E6 +1(q)=E6(q) and E6 −1(q)=2E6(q). We classify, up to conjugacy, all maximal-proper 3-local subgroups of G, that is, all 3-local M<G which are maximal with respect to inclusion among all proper subgroups of G which are 3-local. To this end, we also determine, up to conjugacy, all elementary-abelian 3-subgroups containing Z(G), all extraspecial subgroups containing Z(G), and all cyclic groups of order 9 containing Z(G). These classifications are an important first step towards a classification of the 3-radical subgroups of G, which play a crucial role in many open conjectures in modular representation theory.

UR - http://www.scopus.com/inward/record.url?scp=85042880063&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85042880063&partnerID=8YFLogxK

U2 - 10.1016/j.jpaa.2018.02.018

DO - 10.1016/j.jpaa.2018.02.018

M3 - Article

AN - SCOPUS:85042880063

VL - 222

SP - 4020

EP - 4039

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 12

ER -

_{6}Journal of Pure and Applied Algebra. 2018 Dec 1;222(12):4020-4039. https://doi.org/10.1016/j.jpaa.2018.02.018